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Journal of the London Mathematical Society 1997 56(3):491-503; doi:10.1112/S0024610797005528
© 1997 by London Mathematical Society
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© The London Mathematical Society

A Criterion for the Existence of Transversals of Set Systems

Jerzy Wojciechowski

Department of Mathematics, West Virginia University P.O. Box 6310, Morgantown, West Virginia 26506-6310, USA. E-mail: jerzy{at}math.wvu.edu

Received 20 February 1995. Revision received 1 August 1995.

Nash-Williams [6] formulated a condition that is necessary and sufficient for a countable family A=(Ai)iisinI of sets to have a transversal. In [7] he proved that his criterion applies also when we allow the set I to be arbitrary and require only that {cap}iisinJAi=Ø for any uncountable J{subseteq}I. In this paper, we formulate another criterion of a similar nature, and prove that it is equivalent to the criterion of Nash-Williams for any family u. We also present a self-contained proof that if {cap}iisinJAi=Ø for any uncountable J{subseteq}I, then our condition is necessary and sufficient for the family u to have a transversal.


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